# Advanced Microeconomics/Homogeneous and Homothetic Functions

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## Homogeneous & Homothetic Functions[edit]

For any scalar a function is *homogenous* if **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle f(tx_1,tx_2,\dots,tx_n)=t^kf(x_1,x_2,\dots,x_n) }**
A *homothetic* function is a monotonic transformation of a homogeneous function, if there is a monotonic transformation **Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(z)}**
and a homogenous function such that f can be expressed as

- A function is
*monotone*where**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \forall \;x,y \in \mathbb{R}^{n} \; x \geq y \rightarrow f(x) \geq f(y)}** - Assumption of homotheticity simplifies computation,
- Derived functions have homogeneous properties, doubling prices and income doesn't change demand, demand functions are homogenous of degree 0
- The slope of the MRS is the same along rays through the origin

### Example:[edit]